In the end, New Zealand comfortably defeated Australia, the other formal challenger to stage the tournament, 25 votes to 17.
"I'm over the moon," a beaming Palmer said. "This is about the momentum we've got for women's rugby. With the Black Ferns based in Aotearoa, New Zealand, we pitched it that if you played women's rugby you'd want to come to New Zealand.
"I'm absolutely ecstatic I'm just holding it all in at the moment.
"We always look forward to the women's Rugby World Cup and then we come home and say 'when are we going to have it back here?' This is something we've been wanting for a long time, and the timing is right.
"I spoke with emotion and passion and said this is what it means to us."
New Zealand's pitch to host the six-week tournament received strong Government support from sports minister Grant Robertson, and a personalised video message from Prime Minister Jacinda Ardern, who expressed her delight at the outcome.
"I'm hugely excited the event will be held here. New Zealand is the home of women's rugby, the Black Ferns captured our imaginations again when they won their fifth World Cup title last year, and our entire country will get behind the tournament and all the teams participating in 2021," Ardern said in a statement from the South East Asia Economic Forum.
"This Government is committed to more women and girls getting involved in sport, so we are enthusiastic supporters of bringing this elite women's tournament to New Zealand and inspiring a new generation of women and girls to get involved in rugby."
Matches will be staged in Auckland and Whangarei - Waitakere Stadium, the Northland Events Centre and Albany - from July to August, with the final and, possibly, semifinals to be played at Eden Park.
All 12 teams will experience the same training facilities and hotels used to host the 2011 World Cup and last year's British and Irish Lions tour.
The bid also featured a pledge to run a Pacific Nations tournament alongside the World Cup, with the aim of unleashing that region's untapped female talent.
"We want to do what we can to grow the women's game in the Oceania region," Palmer said. "This will just give us an opportunity to do that.
"Although we are very passionate about women's rugby very few of our fans have seen the Black Ferns play at home against the best in the world.
"I think I was laughing and crying during the men's Rugby World Cup final in 2011 so I will be in a similar state if that's the case.
"This isn't just some superficial thing we decided to do at the last minute. We've been working towards it. We've been putting a lot of investment into the women's game; we've got the Black Ferns contracted now, we've got really good programmes with the provincial teams.
"This is why I think the timing is right."
Palmer has a six-year-old daughter and cannot wait for her to witness the tournament.
"I'm really looking forward to her seeing the best of these female rugby players on our home turf that's why drives me now."
World Rugby Chairman Bill Beaumont said the pitch captured the imagination of voting unions.
"You listen to the presentation and it's the passion for the sport," Beaumont said. "Everybody knows what rugby means to Kiwis. It's in their DNA. That came out in the bid.
"If I was a Kiwi I would think it's coming home."
